C₂×#20276728 #27538135

order	13568
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#20276728

Splittings

Fitting	C₂×C₅₃×#563 ⋊ C₂
Sylow	C₅₃ ⋊ C₂×#2201

Subgroups

	subgroup	G/H
center	C₂²	#16810810
commutator, cosocle	C₂×D₄×C₅₃	C₂⁴
Frattini	C₂×D₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#563	C₂
socle	C₂²×C₅₃	B(4,𝔽₂)
prime radical	C₂×#13133262	C₂²
hypercenter	C₂×#563	D₅₃
hypocenter	C₅₃	C₂×#2201
LCS	G ▷ C₂×D₄×C₅₃ ▷ C₂²×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×D₄ ◁ C₂×#563

1136 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 1136 conjugacy classes of subgroups — too many to draw. Only its 171 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 3392

A minimal faithful permutation representation — 4 generators on 87 points:

(3,6,8,19)(4,10,12,26)(5,14,16,17)(7,20,21,18)(9,22,23,24)(11,25,27,34)(13,28,29,33)(15,31,32,30)(35,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37);(3,5,8,16)(4,9,12,23)(6,17,19,14)(7,15,21,32)(10,24,26,22)(11,13,27,29)(18,31,20,30)(25,33,34,28);(3,4)(5,13)(6,10)(7,11)(8,12)(9,15)(14,28)(16,29)(17,33)(18,25)(19,26)(20,34)(21,27)(22,30)(23,32)(24,31)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85)(86,87);(1,2)

Conjugacy classes & character table

1022 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 323611);